【摘要】：蒙特卡羅(MC)方法作為一種統(tǒng)計(jì)方法,起初主要作為確定論方法的補(bǔ)充使用.但是隨著計(jì)算機(jī)的快速發(fā)展以及對(duì)計(jì)算機(jī)的普及與應(yīng)用打破了這一局面,使得原本耗時(shí)的模擬過(guò)程變得更加快捷,這極大的促進(jìn)了MC方法的發(fā)展.MC方法的應(yīng)用范圍也從起初的核領(lǐng)域延伸到其它領(lǐng)域,現(xiàn)今已成為解決許多物理、工程、金融等領(lǐng)域中問(wèn)題的重要計(jì)算工具,在參數(shù)估計(jì)和可靠性設(shè)計(jì)投資風(fēng)險(xiǎn)和投標(biāo)報(bào)價(jià)生物醫(yī)學(xué)統(tǒng)計(jì)物理學(xué)等方面也有極為廣泛的應(yīng)用,對(duì)于確定性問(wèn)題,也可用MC方法進(jìn)行解決,以及求解各類方程(組)、計(jì)算多重積分、無(wú)窮級(jí)數(shù)等.但是由于MC方法經(jīng)過(guò)隨機(jī)模擬得到的近似值難免會(huì)與所估計(jì)的真值存在偏差,本文針對(duì)該問(wèn)題展開(kāi)研究.首先,論文對(duì)MC方法的收斂性與誤差估計(jì)進(jìn)行了分析,總結(jié)了常用的六種方差縮減的方法:重要抽樣法、控制變量法、對(duì)偶變量法、條件期望法、分層抽樣法、相關(guān)抽樣法.其次,論文討論了常見(jiàn)的方程組、定積分、級(jí)數(shù)問(wèn)題的蒙特卡羅對(duì)偶變量法,通過(guò)建立概率統(tǒng)計(jì)模型、抽樣產(chǎn)生隨機(jī)樣本進(jìn)而獲得確定性問(wèn)題的估計(jì)值,結(jié)果顯示該技巧可有效降低模擬精度且縮短計(jì)算機(jī)運(yùn)行時(shí)間,充分展示了對(duì)偶變量法的高效性.最后,討論線性代數(shù)系統(tǒng)的自適應(yīng)蒙特卡羅求解算法.該方法包括自適應(yīng)重要性抽樣蒙特卡羅和自適應(yīng)相關(guān)抽樣蒙特卡羅,并通過(guò)求解具體算例比較蒙特卡羅方法和確定性方法的求解時(shí)間和算法效率,從而得出自適應(yīng)蒙特卡羅方法可以獲得更快的收斂速度。
[Abstract]:The Monte Carlo (MC) method, as a statistical method, was used primarily as a supplement to the deterministic method. However, with the rapid development of computers and the popularization and application of computers, this situation has been broken, and the time consuming simulation process has become more rapid. This greatly promotes the development of MC methods. The application of MC methods extends from the original nuclear field to other fields. Now, it has become an important computing tool to solve many problems in physics, engineering, finance and so on. It is also widely used in parameter estimation and reliability design investment risk and bid quotation biomedical statistical physics. For deterministic problems, MC method can also be used to solve them. And solve all kinds of equations (groups), calculate multiple integrals, infinite series and so on. But because the approximate value of MC method after random simulation will inevitably deviate from the estimated true value, this paper focuses on this problem. Firstly, the convergence and error estimation of MC method are analyzed, and six commonly used methods of variance reduction are summarized: important sampling method, control variable method, dual variable method, conditional expectation method, stratified sampling method, correlation sampling method. Secondly, the paper discusses the common equations, definite integrals, series of the Monte Carlo dual variable method, through the establishment of probability and statistics model, sampling to generate random samples to obtain the estimated value of the deterministic problem. The results show that this technique can effectively reduce the simulation accuracy and shorten the running time of the computer, which fully demonstrates the efficiency of the dual variable method. Finally, the adaptive Monte Carlo algorithm for linear algebraic systems is discussed. The method consists of adaptive importance sampling Monte Carlo and adaptive correlation sampling Monte Carlo. The solving time and efficiency of Monte Carlo method and deterministic method are compared by solving concrete examples. It is concluded that the adaptive Monte Carlo method can obtain faster convergence rate.
【學(xué)位授予單位】：內(nèi)蒙古工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O212.1

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本文編號(hào)：2130830